Data Manipulation Using R

1 Project proposal

The purpose of this project is to demonstrate my R skills in data manipulation and structuring, as well as an analysis of the presented data. With that in mind, I’ll leave most of the code exposed, but not all so it doesn’t get too repetitive. I must point out that although present in some tables, the “yellow or indigenous” ethnicity will not be presented in the final report, since there are a large number of outliers and that end up biasing the analysis.

During the analysis, I will make several population predictions, as not all tables present data from 2019. For population prediction, I will base myself on the arithmetic method present in the book by Marcos Von Sperling (Von Sperling 2014).

Arithmetic Method

This method assumes a constant growth rate for the following years, based on known data, for example, the population of the last census. Mathematically, it can be represented as follows:

\[\frac{dP}{dt} = k_a\]

where dP/dt represents the population change (P) per unit time (t), and ka is a constant. Considering that P1 is the population of the penultimate census (year t1) and P2, the population of the last census (year t2), we have:

\[\int_{P_2}^{P_1} = k_a\int_{t_1}^{t_2}dt\]

Integrating between the defined limits, we have:

\[P_2-P_1 = k_a(t_2-t_1)\] \[k_a = \frac{P_2-P_1}{t_2-t_1}\]

Using the equation, we arrive at the general expression of the arithmetic method:

\[P=P_2 + k_n(t-t_2)\]

where t represents the year of the projection.

This method admits that the population varies linearly with time and can be used for population forecasting for a short period, from 1 to 5 years. For a forecast for a very long period, the discrepancy with historical reality becomes accentuated, since growth is an unlimited assumption.

To apply this arithmetic method formula, I will use this function below.

formula <- function(P2, t2, P0, t0, t) {
  Ka <- (P2 - P0) / (t2 - t0)
  Pt <- P2 + Ka * (t - t2)
  return(Pt)
}

1.1 Preparing the R Environment

First of all, I’m going to load the packages that I’m going to use to manipulate the data, do the analysis and generate this report.

knitr::opts_chunk$set(cache=TRUE, warning=FALSE, message=FALSE)
library(foreign)
library(lmtest)
library(readxl)
library(writexl)
library(stringi)
library(purrr)
library(tidyverse)
library(knitr)
library(markdown)
library(kableExtra)
library(htmltools)
library(rstatix)
library(emmeans)

2 The INFOPEN

“Infopen is a statistical information system of the Brazilian penitentiary system. The system, updated by the managers of the establishments since 2004, summarizes information about penal establishments and the prison population. In 2014, DEPEN reformulated the methodology used, with a view to modernizing the collection instrument and expand the range of information collected. The treatment of the data allowed a broad diagnosis of the studied reality, but which did not exhaust, in any way, all the possibilities of analysis.”

2.1 Loading Initial Data Step by Step

First I’m going to create a dictionary of states that I’m going to use. I did this in order to standardize the names of the states, as some tables show the full names, others show only the abbreviations of the states.

head(state_dict, 20)

##                Acre             Alagoas               Amapá            Amazonas 
##                "AC"                "AL"                "AP"                "AM" 
##               Bahia               Ceará    Distrito Federal      Espírito Santo 
##                "BA"                "CE"                "DF"                "ES" 
##               Goiás            Maranhão         Mato Grosso  Mato Grosso do Sul 
##                "GO"                "MA"                "MT"                "MS" 
##        Minas Gerais                Pará             Paraíba              Paraná 
##                "MG"                "PA"                "PB"                "PR" 
##          Pernambuco               Piauí      Rio de Janeiro Rio Grande do Norte 
##                "PE"                "PI"                "RJ"                "RN"

Here is the list of columns that I will extract from the INFOPEN tables. I created this list using manipulation with Excel.

Loop through INFOPEN table files and list their names

Match directory to file name

infopen_files <- str_c("INFOPEN/tabelas excel/",infopen_file_name)

Name each vector element

names(infopen_files) <- gsub("\\.xlsx$", "", infopen_file_name)

Apply the read_excel function to each vector element, thus importing all files at once

infopen<- map_df(.x = infopen_files, .f = read_excel, .id = "data") %>%
   select("state" = "UF", date = data, all_of(columns))

2.2 Recognizing the Table

INFOPEN tables present panel data, where each individual is represented more than once.

Each INFOPEN table contains more than 1300 columns and approximately 1500 rows.

jun_2017 <- read_excel("INFOPEN/tabelas excel/jun 2017.xlsx")

Number of columns: 1332 Number of rows: 1514

After analyzing each table, I decided to filter only the most interesting columns for my analysis and the result was a table with 8932 rows and 328 columns.

2.3 Starting to manipulate the dataframe

After grouping all the tables and choosing only the columns that I’m going to use, the next step will be to transform the format from wide to long. Long format facilitates some manipulations, and wide format others. In the course of this analysis I will use both formats.

infopen_2_long_format <-infopen%>%
   gather(variable , quantity, - c(state,date)) %>%
   drop_na()

I will summarize the values so that the repeated lines are removed and the total of each variable is obtained

infopen_3_summary <- infopen_2_long_format %>%
   group_by(state, date, variable) %>%
   mutate(date = gsub("dez", "dec", date))%>% ## I needed to use "dec" so that the program understood that it referred to the month of December
   summarise(prisoners = sum(quantity, na.rm = TRUE)) %>%
   merge(state_region, by = 'state', all.x = TRUE)

2.4 Create the Variables I’m Going to Work With

After summarizing the data and removing the ‘NAs’, it’s time to separate the data into columns that I will use.

infopen_4 <- infopen_3_summary %>%
   rowwise() %>% ## defines the scope of the following operations, to be worked by row and not columns
   filter(!str_detect(variable, "not_informed|not_informed|no information"))%>% ## remove variables that will not be needed
   mutate( ## here I start to define the columns that I will use. I will extract the new columns from the variable column
     gender = case_when(
       str_detect(variable, "(female)") ~ "female",
       str_detect(variable, "(male)") ~ "male",
       TRUE ~ NA_character_),
     variable = gsub("_male|_female|", "", variable), ## at this point I need to remove the gender string to avoid conflicts later in the code
     ethnicity = ifelse(grepl("ethnicity_", variable),
                    sub("ethnicity_", "", variable), NA),
     ethnicity = if_else(ethnicity == "white", "white",
                     if_else(ethnicity %in% c("black", "brown"), "black or brown",
                             ifelse(ethnicity %in% c('yellow','indigenous'), 'yellow or indigenous',NA))),
     level_of_education = ifelse(grepl("level_of_education_", variable),
                                sub("level_of_education_","", variable), NA),
     wage = ifelse(grepl("wage_", variable),
                          sub("wage_","",variable),NA),
     age_range = ifelse(grepl("age_range_", variable),
                           sub("age_range_","",variable),NA))%>%
   mutate_at(vars(-prisoners), as.factor) %>%
   ungroup() %>%
   select("date","region","state", "gender","ethnicity",
          "level_of_education", "age_range", "wage","prisoners") %>% ## I used select() only because I would like to view the columns in that order
   filter(!is.na(gender))

2.5 Generation of the Tables

Regarding the quantity of prisoners, this is the most reliable table because not all detention centers are able to collect all data. So, with the other tables I will work only with the percentage of prisoners in relation to the total and extract the corresponding value from here.

2.5.1 Table with the total prison population

prison_population <-infopen_3_summary %>%
   filter(str_detect(variable, "prison_population"))%>%
   rowwise() %>%
   mutate(
     gender = case_when(
       str_detect(variable, "female") ~ "female",
       str_detect(variable, "male") ~ "male",
       TRUE ~ NA_character_)) %>%
   select(region, state, date, gender, prisoners) %>%
   drop_na() %>%
   ungroup()

prison_population_2_summary <- prison_population %>%
   filter(grepl("^dec", date) | date == "jun 2019") %>% ## after some analysis I decided to use only 1 reference per year, instead of an average of the values.
   group_by(region, state, date) %>%
   mutate(year = str_replace(date, "\\D*(\\d{4}).*", "\\1")) %>% ## removing the first 4 characters from the values in column 'year'
   ungroup() %>%
   group_by(year, region, state) %>%
   summarise(total_prisoners = sum(prisoners)) %>%
   ungroup() %>%
   select(year, region, state, total_prisoners)

I’ll just leave this example of code, because for the creation of the other tables, there is not much difference in relation to the process of this one. What can change are some punctual adjustments, but nothing that deserves mention.

infopen_age_range <-infopen_4 %>%
  select(region,state,date,gender, age_range, prisoners) %>% ## select columns
  mutate(age_range = gsub("_", " ", age_range)) %>% ## remove the "_" to make it easier to read and export to csv
  mutate_at(vars(-prisoners), as.factor) %>% ## convert all columns to factor -prisoners
  drop_na() ## remove NA

infopen_age_range_2_percentage <- infopen_age_range %>%
  filter(grepl("^dec", date) | date == "jun 2019") %>%
  group_by(region, state, date) %>%
  mutate(year = str_replace(date, "\\D*(\\d{4}).*", "\\1"),
         total_prisoners = sum(prisoners, na.rm = TRUE), # total sum of prisoners by region, state, gender and year
         percentage_prisoners = round(((prisoners / total_prisoners) * 100),2),
         state = as.factor(state),
         year = as.character(year)) %>%
  ungroup() %>%
  select(year, region, state, gender, age_range, percentage_prisoners)

infopen_age_range_3_final <- infopen_age_range_2_percentage %>%
  left_join(prison_population_2_summary, by = c("year", "region", "state")) %>%
  mutate(prisoners = round(((percentage_prisoners / 100) * total_prisoners), 0)) %>%
  select(year, region, state, gender, age_range, prisoners)

infopen_age_range_4 <- infopen_age_range_3_final %>%
  group_by(year, age_range) %>%
  summarise(prisoners = sum(prisoners)) %>%
  mutate(year = as.numeric(year))

2.6 INFOPEN Data Visualization

Here are the INFOPEN tables that I will be using. Note that I combined several tables, rearranged the columns and extracted 5 different tables, with panel data.

2.6.1 Prison population

region	state	date	gender	prisoners
Southeast	SP	jun 2019	male	134002
Southeast	SP	dec 2018	male	129255
Southeast	SP	jun 2018	male	124912
Southeast	SP	dec 2017	male	121808
Southeast	SP	jun 2017	male	120332
Southeast	SP	dec 2016	male	119372
Southeast	SP	dec 2016	male	56185
Southeast	SP	jun 2017	male	54000
Southeast	SP	dec 2017	male	51915
Southeast	SP	jun 2018	male	51270

2.6.2 Ethnicity

year	region	state	ethnicity	prisoners
2019	Southeast	SP	black or brown	136560
2018	Southeast	SP	black or brown	132756
2016	Southeast	SP	black or brown	129392
2017	Southeast	SP	black or brown	128026
2016	Southeast	SP	white	100415
2019	Southeast	SP	white	96821
2017	Southeast	SP	white	95929
2018	Southeast	SP	white	94901
2018	Southeast	MG	black or brown	57401
2019	Southeast	MG	black or brown	56458

2.6.3 Education Level

year	region	state	gender	level_of_education	prisoners
2019	Southeast	SP	male	elementary school incomplete	99346
2017	Southeast	SP	male	elementary school incomplete	98797
2018	Southeast	SP	male	elementary school incomplete	97082
2016	Southeast	SP	male	elementary school incomplete	96434
2019	Southeast	SP	male	high school incomplete	47265
2018	Southeast	SP	male	high school incomplete	46693
2017	Southeast	SP	male	high school incomplete	43707
2016	Southeast	SP	male	high school incomplete	42049
2018	Southeast	MG	male	elementary school incomplete	41789
2019	Southeast	MG	male	elementary school incomplete	40967

2.6.4 Age range

year	age_range	prisoners
2016	18 to 24 years old	214624
2018	18 to 24 years old	206467
2019	18 to 24 years old	204917
2017	18 to 24 years old	202629
2019	25 to 29 years old	178737
2018	25 to 29 years old	173891
2016	25 to 29 years old	169540
2017	25 to 29 years old	168851
2019	35 to 45 years old	157032
2018	35 to 45 years old	145968

2.6.5 Pay range

year	region	state	wage	prisoners
2016	Southeast	SP	between 3/4 and 1 monthly minimum wage	230152
2019	Southeast	SP	less than 3/4 of the monthly minimum wage	155026
2018	Southeast	SP	less than 3/4 of the monthly minimum wage	106953
2018	Southeast	SP	does not receive	94809
2017	Southeast	RJ	between 3/4 and 1 monthly minimum wage	51132
2019	Southeast	RJ	does not receive	49260
2019	Southeast	SP	between 3/4 and 1 monthly minimum wage	47639
2019	Southeast	MG	does not receive	46006
2016	Southeast	RJ	between 3/4 and 1 monthly minimum wage	38273
2016	Southeast	MG	does not receive	37727

3 Table IBGE Level of Education

The purpose of this analysis is to compare data from the prison population with data from the IBGE, and make a correlation between them. The data I will use here are part of the National Household Sample Survey Continues (PNADC) and can be found on the IBGE.

Here I start working on the second table that will be used. This single table has several sheets that I will extract and manipulate the data. The table “PNAD_Continua_2018_Educacao.xls” has data regarding the education of the population. There are several pieces of information, including: educational level by region, gender and ethnicity. This table also presents panel data. Yellow and Indigenous are included in the Total

I’m going to skip the table import part and go directly to the dataframe.

Visualization of the PNAD Table

indicator	territorial_level	territorial_opening	variable_1	category_1	variable_2	category_2	2016	2017	2018
População (mil pessoas)	País	Brasil	Sexo	Total	Grupos de idade	Total	204325.470	205999.691	207651.621
População (mil pessoas)	País	Brasil	Sexo	Total	Grupos de idade	0 a 3 anos	10223.229	10141.908	10171.730
População (mil pessoas)	País	Brasil	Sexo	Total	Grupos de idade	4 e 5 anos	5262.563	5268.945	5350.024
População (mil pessoas)	País	Brasil	Sexo	Total	Grupos de idade	6 a 9 anos	11100.370	10962.806	10947.352
População (mil pessoas)	País	Brasil	Sexo	Total	Grupos de idade	10 a 14 anos	15445.672	15363.810	15023.146
População (mil pessoas)	País	Brasil	Sexo	Total	Grupos de idade	15 a 17 anos	10617.588	10426.076	9752.471
População (mil pessoas)	País	Brasil	Sexo	Total	Grupos de idade	18 a 24 anos	22234.284	22727.774	22703.814
População (mil pessoas)	País	Brasil	Sexo	Total	Grupos de idade	25 a 29 anos	15306.030	15138.452	14890.647
População (mil pessoas)	País	Brasil	Sexo	Total	Grupos de idade	30 a 39 anos	32134.993	32462.937	32597.356
População (mil pessoas)	País	Brasil	Sexo	Total	Grupos de idade	40 a 59 anos	52417.780	53172.422	54100.608

This dataframe gathers data from all tabs of the “PNAD_Continua_2018_Educacao.xls” file, there is still a lot of manipulation to be done.

3.1 Data Manipulation

First, I’m going to transpose the data so that I can transform it into long format, just like in the previous model, with the INFOPEN table.

pnad_2_long_format <- pivot_longer(pnad, 8:10,
                                      names_to="year",
                                      values_to = "value",
                                      values_drop_na = TRUE)

I’ll multiply the value in the ‘value’ column by 1000 if the ‘indicator’ column contains the string ‘(mil pessoas)’‘(thousand people)’ and then remove it. Then I’ll create a ‘region’ variable to store the region of each state.

pnad_3_with_regions <- pnad_2_long_format %>%
   mutate(value = ifelse(grepl("(mil pessoas)", indicator), value * 1000, value),
          indicator = gsub("\\s*\\(mil pessoas\\)", "", indicator),
          region = case_when(territorial_opening %in% c("Acre", "Amazonas", "Amapá", "Pará", "Rondônia", "Roraima", "Tocantins") ~ "North",
                             territorial_opening %in% c("Maranhão", "Piauí", "Ceará", "Rio Grande do Norte", "Paraíba", "Pernambuco", "Alagoas", "Sergipe", "Bahia") ~ "Northeast",
                             territorial_opening %in% c("Minas Gerais", "Espírito Santo", "Rio de Janeiro", "São Paulo") ~ "Southeast",
                             territorial_opening %in% c("Paraná", "Santa Catarina", "Rio Grande do Sul") ~ "South",
                             territorial_opening %in% c("Mato Grosso", "Mato Grosso do Sul", "Goiás", "Distrito Federal") ~ "Midwest",
                             TRUE ~ NA_character_))

3.1.1 PNAD table Population aged 18 or over

I’m going to combine this IBGE table with the first INFOPEN table that concerns the prison population, thus also being able to correlate the total number of prisoners with people aged 18 or over, but in a summarized way.

Here we can have an idea of the data present in this table.

## # A tibble: 6 × 4
## # Groups:   year, region [2]
##   year  region  state   total
##   <chr> <chr>   <chr>   <dbl>
## 1 2016  Midwest DF    2884713
## 2 2016  Midwest GO    6712470
## 3 2016  Midwest MS    2614076
## 4 2016  Midwest MT    3299360
## 5 2016  North   AC     826731
## 6 2016  North   AM    3789354

This table has data from 2016 to 2018. The first step will be to calculate the total number of people for each variable in the year 2019, using the arithmetic method described at the beginning of this analysis.

3.2 Using Arithmetic Method to Estimate a Population

I’ll start by transforming this data into a wide format, then I’ll apply the function with the formula and finally return the table to a long format.

population_18_years_or_over_3 <- pivot_wider(population_18_years_and_over_2,
                                             names_from = year,
                                             values_from = total)

population_18_years_or_over_4 <- population_18_years_or_over_3 %>%
  mutate(
    `2019` = round(formula(`2018`,2018,`2016`,2016,2019))
  )

population_18_years_or_over_5 <- pivot_longer(population_18_years_or_over_4, cols = -c(state,region),names_to = "year",values_to = "population") %>%
   mutate(across(-population, as.factor))

I will now combine this table with the INFOPEN prison population.

year	region	state	prisoners	population
2016	Midwest	DF	14958	2884713
2016	Midwest	GO	18626	6712470
2016	Midwest	MS	18320	2614076
2016	Midwest	MT	11642	3299360
2016	North	AC	6100	826731
2016	North	AM	10241	3789354
2016	North	AP	2937	786591
2016	North	PA	14886	8281744
2016	North	RO	12018	1705323
2016	North	RR	2503	476787

3.3 Table PNAD Education Data

I’m going to repeat basically the same process in the table with data on the population. This table, however, considers people aged 14 or over, as can be seen from the indicator.

alphabetization_population <- pnad_3_with_regions %>%
   filter(indicator =="Pessoas de 14 anos ou mais de idade",
          territorial_level == "Unidade da Federação",
          category_1 %in% c("Homem", "Mulher", "Branca", "Preta ou Parda", "Total¹"),
          variable_2 == "Nível de instrução",
          !(category_2 %in% c("Total"))) %>%
  filter(!str_detect(category_1,"Branca|Preta ou Parda|Total¹")) %>% 
   mutate(year = as.numeric(year),
     gender = recode(category_1,
       "Homem" = "male",
       "Mulher" = "female"),
          state = state_dict[as.character(territorial_opening)]) %>%
   select(region, state, gender, level_of_education = category_2, year, total = value) %>%
   drop_na()

Same process to calculate the population in the year 2019.

alphabetization_population_2 <- pivot_wider(alphabetization_population,
                                             names_from = year,
                                             values_from = total)

alphabetization_population_3 <- alphabetization_population_2 %>%
  mutate(
     `2019` = round(formula(`2018`,2018,`2016`,2016,2019))
   )

alphabetization_population_4 <- pivot_longer(alphabetization_population_3, cols = -c(region:level_of_education),names_to = "year",values_to = "population") %>%
   mutate(across(-population, as.factor))

## # A tibble: 6 × 6
##   region state gender level_of_education                        year  population
##   <fct>  <fct> <fct>  <fct>                                     <fct>      <dbl>
## 1 North  RO    male   No education                              2016       51545
## 2 North  RO    male   No education                              2017       45943
## 3 North  RO    male   No education                              2018       44963
## 4 North  RO    male   No education                              2019       41672
## 5 North  RO    male   Incomplete Elementary School (or equival… 2016      265952
## 6 North  RO    male   Incomplete Elementary School (or equival… 2017      277958

As you can imagine, the education level distributions are not standardized. I’m going to use a function to create this pattern between the PNAD table and the INFOPEN table.

standardize_level_of_education <- function(grade) {
   simplified_grade <- gsub(" \\(or equivalent\\)", "", grade)
   recode(simplified_grade,
          "No education" = "illiterate",
          "Incomplete Elementary School" = "elementary school incomplete",
          "Complete Elementary School" = "elementary school complete",
          "Incomplete High School" = "high school incomplete",
          "Complete High School" = "high school complete",
          "Incomplete College/University" = "college or university incomplete",
          "Complete College/University" = "college or university complete",
          "Literacy without regular courses" = "elementary school incomplete
")
}

after running the function on both tables, here is the result:

3.4 PNADC and INFOPEN Standardized Tables

3.4.1 Literacy of the population

year	region	state	gender	level_of_education	population
2016	Midwest	DF	female	college or university complete	326836
2016	Midwest	DF	female	college or university incomplete	81125
2016	Midwest	DF	female	elementary school complete	115600
2016	Midwest	DF	female	elementary school incomplete	243841
2016	Midwest	DF	female	high school complete	361021
2016	Midwest	DF	female	high school incomplete	79367
2016	Midwest	DF	female	illiterate	36913
2016	Midwest	DF	male	college or university complete	269837
2016	Midwest	DF	male	college or university incomplete	80959
2016	Midwest	DF	male	elementary school complete	112742

3.4.2 Literacy of prisoners

year	region	state	gender	level_of_education	prisoners
2016	Midwest	DF	female	college or university complete	10
2016	Midwest	DF	female	college or university incomplete	39
2016	Midwest	DF	female	elementary school complete	34
2016	Midwest	DF	female	elementary school incomplete	317
2016	Midwest	DF	female	high school complete	127
2016	Midwest	DF	female	high school incomplete	118
2016	Midwest	DF	female	illiterate	12
2016	Midwest	DF	male	college or university complete	78
2016	Midwest	DF	male	college or university incomplete	238
2016	Midwest	DF	male	elementary school complete	1496

3.5 Missing data in INFOPEN table

I noticed that after all the standardizations, the tables came back with different number of observations. The infopen_level_of_education_31_standard table has 1509 observations, while alphabetization_population_5 has 1512 observations. I decided to investigate using anti_join and found that the infopen table does not have the observations of the table created below.

## # A tibble: 3 × 6
##   year  region    state gender level_of_education             population
##   <fct> <fct>     <fct> <fct>  <chr>                               <dbl>
## 1 2016  North     RR    female college or university complete      30467
## 2 2017  Northeast MA    female college or university complete     199776
## 3 2016  Northeast SE    female college or university complete     101357

In order not to leave these values blank, I decided to use a simple average of the number of prisoners in other years for each missing observation, use this average as the value and only then combine the PNADC tables with INFOPEN.

Now the INFOPEN table contains 1512 columns, just like the PNAD table, so I can combine them.

3.6 PNADC Table - INFOPEN Level of Education

year	region	state	gender	level_of_education	prisoners	population
2016	Midwest	DF	female	college or university complete	10	326836
2016	Midwest	DF	female	college or university incomplete	39	81125
2016	Midwest	DF	female	elementary school complete	34	115600
2016	Midwest	DF	female	elementary school incomplete	317	243841
2016	Midwest	DF	female	high school complete	127	361021
2016	Midwest	DF	female	high school incomplete	118	79367
2016	Midwest	DF	female	illiterate	12	36913
2016	Midwest	DF	male	college or university complete	78	269837
2016	Midwest	DF	male	college or university incomplete	238	80959
2016	Midwest	DF	male	elementary school complete	1496	112742

4 IBGE Ethnicity Table

This Table has the ethnic percentage distribution of the Brazilian population by state. The file to be worked on here is called “PNADc/Tabela 1.1 DIST PERCET RACA.xls”, and can be found on the IBGE website.

I’ll skip the data reading part as it doesn’t differ at all from the previous tables.

year	state	Total	White	Black	Brown
2018	RO	1747.154	29.47056	6.710325	62.40118
2018	AC	853.023	21.02601	5.346175	72.31678
2018	AM	3921.508	16.80872	2.989718	77.40833
2018	RR	513.466	23.69004	7.674129	60.86162
2018	PA	8472.029	17.83819	8.171171	72.69580
2018	AP	821.545	17.42855	6.996881	74.26081

This table has data from 2012 to 2018. I’m going to use the formula we discussed at the beginning to estimate the population in 2019. I’ll start by transforming the data in the table, as the number present in the total column must still be multiplied by 1000, and the ethnicity values are in percentage in relation to the total.

population_distribution_by_ethnicity_and_region_3 <- population_distribution_by_ethnicity_and_region_2 %>%
   mutate(
     Total = round(Total*1000),
     White = round(White*Total/100),
     Black = round(Black*Total/100),
     Brown = round(Brown*Total/100))

Finally, the table that we will use to match that of INFOPEN

4.1 Ethnic Distribution of the Brazilian Population

year	region	state	ethnicity	population
2016	North	AC	black or brown	674372
2016	North	AC	white	149493
2016	Northeast	AL	black or brown	2470246
2016	Northeast	AL	white	788828
2016	North	AM	black or brown	3034052
2016	North	AM	white	679764
2016	North	AP	black or brown	619211
2016	North	AP	white	162565
2016	Northeast	BA	black or brown	11944048
2016	Northeast	BA	white	2608310

4.2 Missing Observations in the INFOPEN Table

After manipulating the data from this IBGE table, I will combine it with the INFOPEN table, in order to correlate the total number of prisoners and the population for each variable. However, when combining the dataframes, I discovered that there are missing observations in the infopen_etnia_3_final table because it has fewer rows than the PNAD table. I will use anti-join to find them and linear regression to calculate them

These are the missing observations in the INFOPEN table

head(difference_infopen_population_ethnicity)

## # A tibble: 2 × 4
## # Groups:   year, state, region [1]
##    year region    state ethnicity     
##   <dbl> <chr>     <chr> <chr>         
## 1  2019 Northeast SE    black or brown
## 2  2019 Northeast SE    white

After some calculations, I arrived at this result of predicting prisoners and each observation:

head(predictions_2019_infopen_ethnicity)

## # A tibble: 2 × 5
## # Groups:   region, state, ethnicity [2]
##   region    state ethnicity      prisoners  year
##   <chr>     <chr> <chr>              <dbl> <dbl>
## 1 Northeast SE    black or brown      5124  2019
## 2 Northeast SE    white                457  2019

Now it is enough to combine the tables with data on the ethnicity of the total Brazilian population with the prison population and then we will arrive at this table:

4.3 PNADC Table - INFOPEN Ethnicity

year	region	state	ethnicity	prisoners	population
2016	Midwest	DF	black or brown	12357	1753738
2016	Midwest	DF	white	2513	1113132
2016	Midwest	GO	black or brown	14286	4253442
2016	Midwest	GO	white	4266	2426334
2016	Midwest	MS	black or brown	12555	1452699
2016	Midwest	MS	white	5481	1130769
2016	Midwest	MT	black or brown	8900	2211143
2016	Midwest	MT	white	2655	1076121
2016	North	AC	black or brown	5306	674372
2016	North	AC	white	506	149493

5 IBGE Age Range Table

The file to be worked on here is called “Tabela 1.2 DIST POP ETARIA.xls”, and can be found on the IBGE website.

This table has data on the age group distribution of the population by ethnicity. Indigenous people, Asians and people with no declaration of color or race are included in the total.

age_range	total	cv_total	white	cv_white	black_brown	cv_black_brown	proportion_white	cv_proportion_white	proportion_black_brown	cv_proportion_black_brown	year
0 to 4 years old	13124.30	0.88427169197711608	6076.469	1.451866	6939.813	1.155815	46.29937	1.021138	52.87758	0.8912887	2018
5 to 9 years old	13645.48	0.85759498658148692	5629.184	1.409402	7899.687	1.121951	41.25310	1.074214	57.89233	0.7720677	2018
10 to 14 years old	14923.04	0.77302798278891072	5756.714	1.359373	9039.822	1.043748	38.57603	1.111292	60.57629	0.7071651	2018
15 to 19 years old	16442.07	0.75758111621423652	6010.282	1.296194	10293.966	1.031182	36.55429	1.105113	62.60748	0.6485969	2018
20 to 24 years old	16048.18	0.75135656586436339	6348.342	1.308184	9541.958	1.019381	39.55802	1.057401	59.45820	0.7094211	2018
25 to 29 years old	15006.06	0.83216738688741987	6160.098	1.441365	8686.275	1.068971	41.05073	1.082387	57.88510	0.7691143	2018
30 to 34 years old	16071.92	0.84090771846315848	6600.026	1.448793	9274.408	1.043940	41.06559	1.057334	57.70568	0.7619637	2018
35 to 39 years old	16905.90	0.82142719564900768	7026.611	1.517426	9688.656	1.026723	41.56308	1.113020	57.30932	0.8102016	2018
40 to 44 years old	15186.72	0.78765297599262696	6390.305	1.356850	8621.957	1.049632	42.07823	1.035301	56.77299	0.7678529	2018
45 to 49 years old	13519.07	0.83775955165508598	5857.937	1.412991	7506.854	1.127709	43.33091	1.061985	55.52788	0.8343097	2018

Like the previous table, this one only has data from 2012 to 2018, so I will use the same formula to predict the population in 2019.

5.1 Standardization of Age Ranges

As you can see below, although we now have data from 2016 to 2019, the age groups are not exactly the same as those from INFOPEN, but they are very close.

age_range	year	population
25 to 29 years old	2018	15006064
25 to 29 years old	2017	15361348
25 to 29 years old	2016	15417271
25 to 29 years old	2019	14834679
30 to 34 years old	2018	16071915
30 to 34 years old	2017	16187827
30 to 34 years old	2016	16565123
30 to 34 years old	2019	16054589
35 to 39 years old	2018	16905898
35 to 39 years old	2017	16504662

What I’m going to do is create a function that adjusts the age groups to be in accordance with those of INFOPEN

adjust_age_range <- function(range) {
   if (range %in% c("25 to 29 years old")) {
     return("25 to 29 years old")
   } else if (range %in% c("30 to 34 years old")) {
     return("30 to 34 years old")
   } else if (range %in% c("35 to 39 years old", "40 to 44 years old")) {
     return("35 to 45 years old")
   } else if (range %in% c("45 to 49 years old", "50 to 54 years old", "55 to 59 years old")) {
     return("46 to 60 years old")
   } else if (range %in% c("60 to 64 years old", "65 to 69 years old")) {
     return("61 to 70 years old")
   } else if (range %in% c("70 to 74 years old", "75 to 79 years old", "80 years old and over")) {
     return("over 70 years old")
   } else {
     return(NA)
   }
}

Here you can have a visualization of the table that I have until then.

year	age_range	population
2016	25 to 29 years old	15417271
2016	30 to 34 years old	16565123
2016	35 to 45 years old	30697363
2016	46 to 60 years old	37633650
2016	61 to 70 years old	16444403
2016	over 70 years old	12997738
2017	25 to 29 years old	15361348
2017	30 to 34 years old	16187827
2017	35 to 45 years old	31462107
2017	46 to 60 years old	37959837

It may be noted that I do not have the 18 to 24 year old population. I’m going to extract this age range from another PNAD table, which we’ve worked on before.

population_18_to_24_years <- pnad_4_population_age %>%
   filter(grepl("18 a 24 anos", age_group),
          grepl("Total¹", ethnicity)) %>%
   select(-gender) %>% # I will remove gender to remove duplicates (because I have gender and ethnicity)
   rename(age_range = age_group,
          population = value)%>%
   group_by(year, age_range, ethnicity) %>%
   summarise(population = sum(population)) %>%
   select(-ethnicity) %>% # finally I remove the ethnicity column that only contains 'Total'
   drop_na()

As previously demonstrated, I will use arithmetic to predict the population in 2019.

5.2 Distribution of the Age Range of the Brazilian Population

Finally, the table with all age groups equal to INFOPEN

age_range	year	population
18 to 24 years old	2016	22234284
25 to 29 years old	2016	15417271
30 to 34 years old	2016	16565123
35 to 45 years old	2016	30697363
46 to 60 years old	2016	37633650
61 to 70 years old	2016	16444403
over 70 years old	2016	12997738
18 to 24 years old	2017	22727774
25 to 29 years old	2017	15361348
30 to 34 years old	2017	16187827

Now it remains only to combine the age range tables.

population_infopen_age_range <- as.data.frame(left_join(infopen_age_range_4,
                                             population_age_range,
                                             by = join_by(year, age_range))) %>%
   mutate(age_range = as.factor(age_range))

5.3 PNADC Table - INFOPEN by Age Group

year	age_range	prisoners	population
2016	18 to 24 years old	214624	22234284
2016	25 to 29 years old	169540	15417271
2016	30 to 34 years old	127187	16565123
2016	35 to 45 years old	133083	30697363
2016	46 to 60 years old	49033	37633650
2016	61 to 70 years old	7527	16444403
2016	over 70 years old	1376	12997738
2017	18 to 24 years old	202629	22727774
2017	25 to 29 years old	168851	15361348
2017	30 to 34 years old	130889	16187827

6 IBGE Income Table

6.1 Total population aged 14 and over.

I need this table with the general population over 14 years old, as the IBGE income table only considers this age group. The table on education, which we have already used, considers this range of the population.

6.1.1 Table of Total Population aged 14 or Over.

region	state	year	population
Midwest	DF	2016	2335338
Midwest	DF	2017	2391985
Midwest	DF	2018	2452741
Midwest	DF	2019	2511443
Midwest	GO	2016	5384235
Midwest	GO	2017	5505555
Midwest	GO	2018	5599123
Midwest	GO	2019	5706569
Midwest	MS	2016	2073359
Midwest	MS	2017	2109297

6.2 Data Explanation

This table has the income distribution of the Brazilian population. The IBGE itself released an informative on the income distribution of the Brazilian population between 2012 and 2019.

I will only work with a fraction of the data available in this table: income usually received, at average prices and only for people aged 14 and over. According to the IBGE, usual income is defined as follows:

” The usual income consists of the monthly income received by employees, employers and self-employed workers, without extraordinary increases or sporadic discounts. For the employee, the monthly income usually received excludes all installments that are not continuous (annual bonus, salary late, overtime, annual profit sharing, 13th salary, 14th salary, salary advance, etc.) and does not consider occasional discounts (absences, part of the 13th salary anticipated, possible damage caused to the enterprise, etc.).

If the income received from an employee, self-employed worker and employer is variable, the usual income is considered to be the average income received by the person in the period in which he/she carried out the declared work in the reference week. When remuneration varies depending on the period or season of the year, the monthly income that the person usually earns in that seasonal period is considered.” see it

6.3 Data Exploration

This table is very simple. In the ‘class’ column, we have the percentage class of people by income, and in the other columns, the usual income of this class of people.

I will use a table already present in the IBGE report to better exemplify the use of the table

In the first line ‘2012’, in the column ‘More than 80% up to 90%’, we have the value 3 351, which represents a monthly income of R$ 3,351.00. That is, 90% of Brazilians receive up to this amount, only 10% receive more than that.

6.4 Data Manipulating

So that I can standardize the PNAD and INFOPEN income tables, I will need an adjustment according to the minimum wage. For this I will create a table with the values of the years 2016 to 2019.

year <- c(2019, 2018, 2017, 2016)
minimum_salary <- c(998.00, 954.00, 937.00, 880.00)
minimum_salary_2016_to_2019 <- data.frame(year, minimum_salary)

This table considers only the percentage of people with some income. The IBGE considers unemployed people who are looking for work during the sample period. However, it does not consider people without income who were not looking for a job as unemployed. This portion of people without income is the one we are going to deal with here.

To extract this data, I will use the same table already used, which also has this information. The ‘PNAD_Continua_2019_Rendimento_de_Todas_as_Fontes’ table has the “Percentage of people with income” as an indicator, so I will extract ‘100%’ from this value and obtain the percentage of people without income.

percentage_employed <-population_income %>%
    filter(`Abertura geográfica` == "Brasil",
         Tipo == "Valor",
         sub.classe %in% c("Todas as fontes¹"   ),
         ind == "Percentual de pessoas com rendimento, na população residente") %>% 
  select(sub.classe, '2016','2017','2018','2019') %>% 
  unique()

percentage_employed_2_long_format <- pivot_longer(percentage_employed,
                                                  cols = c("2016", "2017","2018", "2019"),
                                                  names_to="year", 
                                                values_to="population_percentage_with_income")

percentage_pp_without_income <- percentage_employed_2_long_format %>%
   mutate(population_without_income = (100 - population_percentage_with_income),
          income = 0,
          wage_range = "does not receive",
          year = as.factor(year))

no_income <- left_join(percentage_pp_without_income, population_total_14_years_old_or_over, by = join_by(year)) %>%
   mutate(population = (population_without_income*population)/100,) %>%
   group_by(year, wage_range) %>%
   summarise(population = sum(population)) %>%
   select(year, wage_range, population)

Before merging the tables, I still need to standardize the variables.

# Creating the new column with the categories
population_wage_range_3 <- population_wage_range_2 %>%
   mutate(wage_range = case_when(
     income >= minimum_salary & income < 2 * minimum_salary ~ "between 1 and 2 monthly minimum wages",
     income >= 3/4 * minimum_salary & income < minimum_salary ~ "between 3/4 and 1 monthly minimum wage",
     income >= 2 * minimum_salary ~ "over 2 monthly minimum wages",
     income > 0 & income< 3/4 * minimum_salary ~ "less than 3/4 of the monthly minimum wage"
   )) %>%
   select(sub.class, year, population_percentage, income,wage_range)


population_wage_range_4 <-merge(population_wage_range_3,
                                 population_total_14_years_old_or_over, by = "year") %>%
   mutate(paid_population = round((population * population_percentage)/100)) %>%
   select(year, wage_range, paid_population) %>%
   rename(population = paid_population)

population_wage_range_5 <- population_wage_range_4 %>%
   group_by(year, wage_range) %>%
   summarise(population = sum(population)) %>%
   mutate(year = as.factor(year))

# Set the correct order of yields
income_range_order <- c("does not receive",
                                "less than 3/4 of the monthly minimum wage",
                                "between 3/4 and 1 monthly minimum wage",
                                "between 1 and 2 monthly minimum wages",
                                "over 2 monthly minimum wages")

population_remuneration_6 <- rbind(population_wage_range_5, no_income)%>%
   mutate(wage_range = as.factor(wage_range),
          wage_range = factor(wage_range, levels = income_range_order))

Finally, an overview of the distribution of income in the country:

6.5 Population distribution with and without income

year	wage_range	population
2016	does not receive	64375850
2017	does not receive	65298473
2018	does not receive	64819488
2019	does not receive	63997726
2016	less than 3/4 of the monthly minimum wage	16549064
2017	less than 3/4 of the monthly minimum wage	33486396
2018	less than 3/4 of the monthly minimum wage	33848303
2019	less than 3/4 of the monthly minimum wage	34223380
2016	between 3/4 and 1 monthly minimum wage	33098128
2017	between 3/4 and 1 monthly minimum wage	16743198

Before merging the dataframes, I noticed that the Infopen table does not have data on prisoner pay in 2017 for the state of Sao Paulo. I will use linear interpolation (Larson 1988)to predict this data.

# Subset of data for the state of São Paulo
infopen_sp <- infopen_wage_3_final[infopen_wage_3_final$state == 'SP', ]

Function to predict the missing data on the remuneration of prisoners in Sao Paulo in 2017

predict_mv <- function(year, prisoners) {
   complete_cases <- !is.na(prisoners)
   approx(x = as.numeric(year[complete_cases]),
          y = prisoners[complete_cases],
          xout = as.numeric(year))$y
}

Apply the function for each gender and compensation combination

infopen_sp <- infopen_sp %>%
   group_by(wage) %>%
   mutate(prisoners = round(predict_mv(year, prisoners))) %>%
   ungroup()

Replace the original data for Sao Paulo with the new populated data

infopen_wage_3_final[infopen_wage_3_final$state == 'SP', ] <- infopen_sp

Finally the final table the estimated amount of prisoners by remuneration

infopen_wage_4 <- infopen_wage_3_final %>%
   group_by(year, wage) %>%
   mutate(year = as.factor(year)) %>%
   summarise(prisoners = sum(prisoners)) %>% 
    rename(wage_range = wage)

After all the manipulations, I can finally combine the tables.

6.6 PNADC Table - INFOPEN wage range

year	wage_range	prisoners	population
2016	between 1 and 2 monthly minimum wages	45996	66196256
2016	between 3/4 and 1 monthly minimum wage	383948	33098128
2016	does not receive	174533	64375850
2016	less than 3/4 of the monthly minimum wage	81645	16549064
2016	over 2 monthly minimum wages	16286	49647190
2017	between 1 and 2 monthly minimum wages	40861	66972792
2017	between 3/4 and 1 monthly minimum wage	289068	16743198
2017	does not receive	251953	65298473
2017	less than 3/4 of the monthly minimum wage	120993	33486396
2017	over 2 monthly minimum wages	2858	50229594

7 Presentation of Collected and Manipulated Data:

7.1 PNAD Table - INFOPEN Total

year	region	state	prisoners	population
2016	Midwest	DF	14958	2884713
2016	Midwest	GO	18626	6712470
2016	Midwest	MS	18320	2614076
2016	Midwest	MT	11642	3299360
2016	North	AC	6100	826731
2016	North	AM	10241	3789354
2016	North	AP	2937	786591
2016	North	PA	14886	8281744
2016	North	RO	12018	1705323
2016	North	RR	2503	476787

7.2 PNADC Table - INFOPEN Level of Education

year	region	state	gender	level_of_education	prisoners	population
2016	Midwest	DF	female	college or university complete	10	326836
2016	Midwest	DF	female	college or university incomplete	39	81125
2016	Midwest	DF	female	elementary school complete	34	115600
2016	Midwest	DF	female	elementary school incomplete	317	243841
2016	Midwest	DF	female	high school complete	127	361021
2016	Midwest	DF	female	high school incomplete	118	79367
2016	Midwest	DF	female	illiterate	12	36913
2016	Midwest	DF	male	college or university complete	78	269837
2016	Midwest	DF	male	college or university incomplete	238	80959
2016	Midwest	DF	male	elementary school complete	1496	112742

7.3 PNADC Table - INFOPEN Ethnicity

year	region	state	ethnicity	prisoners	population
2016	Midwest	DF	black or brown	12357	1753738
2016	Midwest	DF	white	2513	1113132
2016	Midwest	GO	black or brown	14286	4253442
2016	Midwest	GO	white	4266	2426334
2016	Midwest	MS	black or brown	12555	1452699
2016	Midwest	MS	white	5481	1130769
2016	Midwest	MT	black or brown	8900	2211143
2016	Midwest	MT	white	2655	1076121
2016	North	AC	black or brown	5306	674372
2016	North	AC	white	506	149493

7.4 PNADC table - INFOPEN Age Range

7.5 PNADC Table - INFOPEN Wage Range

year	region	state	gender	level_of_education	prisoners	population
2016	Midwest	DF	female	college or university complete	10	326836
2016	Midwest	DF	female	college or university incomplete	39	81125
2016	Midwest	DF	female	elementary school complete	34	115600
2016	Midwest	DF	female	elementary school incomplete	317	243841
2016	Midwest	DF	female	high school complete	127	361021
2016	Midwest	DF	female	high school incomplete	118	79367
2016	Midwest	DF	female	illiterate	12	36913
2016	Midwest	DF	male	college or university complete	78	269837
2016	Midwest	DF	male	college or university incomplete	238	80959
2016	Midwest	DF	male	elementary school complete	1496	112742

8 Analyzes and Correlations.

Most of the project proposal has already been passed. From this point on, I focus more on presenting some correlations found in the tables we set up and present a little above.

I’ll start by creating a column that relates the number of prisoners to the total population, and then I’ll plot some graphs that illustrate the correlation between each variable

8.1 Ethnicity Dataframe

The variation in the percentage of prisoners in relation to population by ethnicity shows that in practically all states, there is a higher incidence of brown and black prisoners compared to the population. This does not indicate causality, as there are other factors that could influence individuals to commit crimes and end up in jail. However, it is a fact that requires further investigation. It would be ideal to assess additional variables, such as potential racism within the judiciary, as well as education and income, as we are doing here. In the following graphs, I present some correlations between these variables.

8.2 Age Range Dataframe

Here we can observe a decrease in the rate of prisoners in the age group of 18 to 24 years over time. This can happen for several reasons, among them the aging of the prison population. However, if we take all other age groups, we have a considerable increase in the number of prisoners in what we call “working age”, which comprises the population up to 61 years of age. It’s really frustrating to realize that our “Bill Gates”, “Zuckerbergs”, and “Elon Musks” are behind bars. The population that should be in college is trapped, by numerous factors, among which the young age, along with low education and lack of money. It would really be a dream not to have so many young people arrested.

8.3 Education Level Dataframe

This table shows the distribution of prisoners by level of education. Clearly, the key turns in incomplete secondary education, since from then on, the percentage of prisoners over the population drops drastically. It is no longer a mystery that a population with low education usually has a high degree of violence as a response, take countries like norway, netherlands and japan for example where there are very few prisoners, and compare the level of access to higher education with that of Brazil.

8.4 Incomes Dataframe

Here we can observe that most of the prison population are people who receive up to 1 monthly minimum wage. This amount between 3/4 and 1 monthly minimum wage includes several people who receive government aid such as “Bolsa Família” or others. They cannot be, according to the IBGE, classified as without income or unemployed.

The Inter-Union Department of Statistics and Socioeconomic Studies (Dieese), monthly publishes the value of the cost of the Basic Food Basket which, according to the body, would be “sufficient for the sustenance and well-being of an adult worker, containing balanced amounts of protein, calories, iron calcium and phosphorus.(DIEESE 2019a)”. In 2019, the average value of the national Food Parcel was BRL 422.19, which represents almost half the minimum wage at the time (BRL 998.00). ### Minimum Wage Required

The Constitution of Brazil, enacted in October 1988, mandates that the minimum wage should be a legally defined and uniform amount nationwide. It should be sufficient to meet the basic needs of a worker and their family, including housing, food, education, health, leisure, clothing, hygiene, transportation, and social security. The Constitution also requires periodic adjustments to maintain the purchasing power of the minimum wage (Article 7, IV of the Federal Constitution of Brazil).

DIEESE, when calculating the Minimum Necessary Wage, adheres to these constitutional provisions. They base their calculations on Decree Law No. 399, which stipulates that the cost of food for an adult worker should not be lower than the expense of the Basic Food Basket.

In these calculations, DIEESE considers a family model consisting of two adults and two children, assuming that the children’s consumption is equivalent to that of an adult.

The method for calculating a family’s food expenses begins with the cost of the most expensive Basic Food Basket among the 27 Brazilian capitals, which is then multiplied by three.

(DIEESE 2019b) conducted the Family Budget Survey (POF) in São Paulo during the period of 94/95. The results revealed that food accounted for 35.71% of the expenses of families in the lowest income bracket. By comparing the cost of food for a family (the most expensive basket multiplied by three) with the proportion of these families’ budget allocated to food (35.71%), it is possible to calculate the total budget required to cover other expenses such as housing, clothing, transportation, and more.

Therefore, the formula for calculating the Minimum Required Wage can be summarized as follows: \[F.F.C. = 3(CC)\]

\[\frac{F.F.C.}{X} = \frac{0.3571}{1.00}\] Using rule of 3, we have: \[F.F.C. = X(0.3571)\] so: \[ X = \frac{F.F.C.}{0.3571} \] Where: F.F.C. = Family Food Cost and C.C. = Cost of the highest value Food Parcel

The Necessary Minimum Wage, which is calculated monthly as an assessment of what the current minimum wage should be, also serves as a tool that workers’ unions use to expose the violation of the constitutional principle that defines the parameters for determining the lowest allowable wage. in the country.

We have below the value of what would be the ideal salary of the worker, provided for by law, to cover all monthly costs of his residence.

Month	Minimum Wage	Necessary Wage
December	998	4342.57
November	998	4021.39
October	998	3978.63
September	998	3980.82
August	998	4044.58
July	998	4143.55
June	998	4214.62
May	998	4259.90
April	998	4385.75
March	998	4277.04

Through these data, we can see the discrepancy between the minimum values and those necessary for the maintenance of the home in Brazil. The demand for political and economic reforms in Brazil is not recent. Still in the 1970s, the group “Legião Urbana” already raised protest with the song “What country is this?”. In the following decade, we can see the singer Cazuza protesting the song “Brasil”, which clearly would denounce the nation’s poverty.

The prison population says a lot about the country. Young, poor, low-educated and dark-skinned people are at the top of the statistics, which signals an omission on the part of the government. The Penal Code, in its article 135, describes the crime of omission of help, which consists of the attitude of failing to help people in a vulnerable situation, such as abandoned or lost children, disabled people, with injuries, or in a situation of risk or danger. For that reason, the government should also be behind bars.

Biography

DIEESE. 2019a. “Metodologia Da Cesta Básica de Alimentos.” https://www.dieese.org.br/metodologia/metodologiaCestaBasica/?page=1.

———. 2019b. “Pesquisa Nacional Da Cesta Básica de Alimentos.” https://www.dieese.org.br/analisecestabasica/salarioMinimo.html#2019.

Larson, David R. 1988. “Reflexivity, Algebraic Reflexivity and Linear Interpolation.” American Journal of Mathematics 110 (2): 283–99.

Von Sperling, M. 2014. Princípios Do Tratamento Biológico de Águas Residuárias. Vol. 1. Introdução à Qualidade Das Águas e Ao Tratamento de Esgotos. 4th ed. Editora UFMG.